36.6.9 problem 9

Internal problem ID [6370]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 9
Date solved : Monday, January 27, 2025 at 01:58:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-2 x \right ) y^{\prime \prime }+2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 46

Order:=6; 
dsolve((x^2-2*x)*diff(y(x),x$2)+2*y(x)=0,y(x),type='series',x=1);
 

\[ y = \left (1+\left (x -1\right )^{2}+\frac {\left (x -1\right )^{4}}{3}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{3}+\frac {2 \left (x -1\right )^{5}}{15}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 47

AsymptoticDSolveValue[(x^2-2*x)*D[y[x],{x,2}]+2*y[x]==0,y[x],{x,1,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{3} (x-1)^4+(x-1)^2+1\right )+c_2 \left (\frac {2}{15} (x-1)^5+\frac {1}{3} (x-1)^3+x-1\right ) \]